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 A077058 Minimal positive solution a(n) of Diophantine equation b(n)^2 - b(n)*a(n) - G(n)*a(n)^2 = +1 or -1 with G(n) := A078358(n). The companion sequence is b(n)=A077057(n). 4
 1, 1, 2, 1, 1, 8, 2, 10, 1, 1, 40, 5, 2, 3, 250, 1, 1, 106, 3, 1138, 2, 8, 25, 146, 1, 1, 2968, 15, 298, 16, 2, 5, 352, 17, 1856, 1, 1, 9384, 97, 10, 8, 253970, 2, 72664, 3, 6440, 5, 521904, 1, 1, 3034, 5, 9148450, 3, 1084152, 117, 2, 45, 746, 10, 88, 157, 126890, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS This equation can also be written as (2*b(n)-a(n))^2 - D(n)*a(n)^2 = +4 or -4 with D(n) := A077425(n)=1+4*G(n). This is from Perron's table (see reference p. 108, for n = 1..28) which gives the minimal x,y values which solve the above mentioned Diophantine equations. For Pell equation x^2 - D*y^2 = +4, see A077428 and A078355. For Pell equation x^2 - D*y^2 = -4, see A078356 and A078357. REFERENCES O. Perron, "Die Lehre von den Kettenbruechen, Bd.I", Teubner, 1954, 1957 (Sec. 30, Satz 3.35, p. 109 and table p. 108). LINKS Table of n, a(n) for n=1..65. MATHEMATICA g[n_] := Ceiling[ Sqrt[n] ] + n - 1; r[n_] := Reduce[an > 0 && (bn^2 - bn *an - g[n]*an^2 == 1 || bn^2 - bn *an - g[n]*an^2 == - 1), {an, bn}, Integers] /. C -> c; ab[n_] := DeleteCases[ Flatten[ Table[{an, bn} /. {ToRules[r[n]]} // Simplify, {c[1], 0, 1}] , 1] , an | bn]; a[n_] := a[n] = Min[ ab[n][[All, 1]] ]; Table[ Print[{n, a[n]}]; a[n], {n, 1, 65}] (* Jean-François Alcover, Oct 03 2012 *) PROG (PARI) forstep(D=1, 1000, 4, if(issquare(D), next); u=bnfinit(x^2-D).fu[1]; k=1; while( denominator(t=polcoeff(lift(u^k), 1)*2)>1, k++); print1(abs(t), ", "); ) \\ Max Alekseyev, Feb 06 2010 CROSSREFS Sequence in context: A134470 A342992 A119418 * A053373 A297733 A255812 Adjacent sequences: A077055 A077056 A077057 * A077059 A077060 A077061 KEYWORD nonn AUTHOR Wolfdieter Lang, Nov 29 2002 EXTENSIONS More terms from Max Alekseyev, Feb 06 2010 STATUS approved

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Last modified February 23 11:40 EST 2024. Contains 370283 sequences. (Running on oeis4.)